Dave realised that, although
there was no obvious winner, most of
the good ones were subsets of a particular 31 tone
periodicity block which
was very
even
melodically. It was a 72-EDO
tempering
of a 31-tone planar
microtemperament that he had created in December 1999 in collaboration with
Paul Erlich and Carl Lumma.

Paul suggested that Blackjack would be an excellent name for the latter.

Then Dave Keenan performed a computer-assisted search for
other single-chain 11-limit generators which either gave no fewer
hexads
per note at greater accuracy,
or gave more 11-limit quasi-just hexads per note.
He repeated the search at the 7-limit
(for tetrads
instead of hexads).
There were none. In fact no other
generator even came close.

Any tuning with a generator in the range between 116.1 cents
(= 31-EDO) and 117.8 cents (= 0.7 cents larger than 41-EDO) has
MIRACLE
properties, but that of 72-EDO comes closest to
the optimal generators calculated by a number of methods
(including root mean square and maximal absolute).
There is no single optimum MIRACLE generator; many different
kinds of optimum are within 0.15 cent either side of 72-EDO.

(At the time of its rediscovery in April 2001,
Dave, Paul, and Carl were all unaware that
George Secor had published this generator -- but not the MOS scales
-- in 1975, in his article
A
New Look at the Partch Monophonic Fabric, originally
appearing in XenharmonikÃ´n 3. In honor of George,
this interval was subsequently named the
secor.)

Graham Breed has pointed out that
10- and 11-EDO,
whose generators lie a bit outside this range, exhibit some
of the same melodic properties as the MIRACLE scales.
These facts:

have led Graham to advocate the use of a
decimal-based
notational system for scales in the MIRACLE family.
(See also
Graham's decimal page.)

The Blackjack
generator is also nearly identical
to an interval the size of 7 degrees of
72-EDO
[= exactly 116 & 2/3 cents, or 1 & 1/6
Semitones].
Thus, it can be represented quite accurately as a
21-out-of 72-EDO tuning. Because of the many useful
properties associated with 72-EDO, this greatly simplifies
many aspects of the presentation of the Blackjack scale,
such as its notation or its diagramming on a
lattice.

I will present the Blackjack scale here as 21-out-of-72-EDO.
In the 72edo version of blackjack, L
is 5 degrees of 72edo and s is 2 degree, so L = 2.5s.

It was decided by Dave and others that "D" should be the
reference note for the system, because the layout of notes on the standard
Halberstadt keyboard
is symmetrical around "D".
But I used "C" as the reference in my examples here.

Here is a graph showing the pitch-height of the notes in
this scale, within one "octave". Each note is labeled
with both my ASCII adaptation of the Sims/Herf 72-EDO
notation, and its
Semitone
value.

The reference pitch ("0") occurs on the ledger line between staves,
and each successive ascending space and line represents the next
secor-sized
positive
generator
in the decimal series, and thus the next
cardinality in Breed's notation: 1, 2, 3, ... 9, and when
that set is exhausted, the next (and last) in the series
of generators, 0v, is notated as the "0" reference pitch,
an "8ve" higher, accompanied by the v symbol which indicates
lowering by a quomma. Descending from
the higher "0" ledger-line is the negative series of generators,
9^, 8^, 7^, ... 0^, notated on the same line or space as
the namesake cardinality, but accompanied by a ^ symbol
to indicate raising by a quomma.

Here is an
interval
matrix chart of all
dyadic
intervals available in the 21-out-of-72 Blackjack scale,
with the pitches labeled with
their 21-tone Blackjack degree numbers and Semitone values.
All interval sizes are shown in Semitones.

(Thanks to Paul Erlich for color-coding the intervals
according to the
odd-
limits
of the
JI
intervals they represent.)

Paul also made a 7-limit
lattice
diagram showing the
periodicity
blocks implied by the Blackjack tuning, as well
as illustrating the many harmonic structures implied
by this scale. I have adapted it here to my own
ASCII 72-EDO notation. The Blackjack scale is
wafso-just
with respect to this lattice.

Here is a
mapping which I designed, placing 72-EDO onto
the fingerboard/keyboard of a Starr Labs
Ztar
instrument, showing all 72-EDO degrees and their
ASCII Monzo notation. The placement of black and white keys
reflects the association of various 72-EDO notes with those
in 12-EDO as they appear on a regular Halberstadt piano keyboard.
The Blackjack notes are shown in orange.

Notice how the placement of the Blackjack notes in the above
mapping shifts upward by one key as one travels to the right,
because the generator of 7/72 is one more than the 6/72 steps
in each column of the keymap.

In the mapping below, I adapted the Starr Labs
Zboard
keyboard so that each column is 7 steps high, thereby making all the
Blackjack notes adjacent.

Below is an applet which shows the 5-limit representation
of Graham's chord progression. Mouse-over the chord-number
(without clicking) to
see a lattice of that chord in red. Commatic equivalents are
shown in purple. Note that Graham's chords all imply a 7-limit
harmony, which is shown in its closest 5-limit approximation (225:128
above the "root" of the chord) here.

For the benefit of those wishing to map this family of
tunings to a standard 12-tone
Halberstadt
keyboard, Paul Erlich devised an interesting 12-tone subset of Blackjack,
presented in
Tuning
List post 22532 from Sat May 12, 2001 8:35 am.

Dave also created
this diagram of a color-coded design for mapping
the full Blackjack scale to the Halberstadt keyboard.

As stated above, 31edo is the tuning which provides
the lower limit of the blackjack generator, so those
who work in 31edo may easily form blackjack as a subset
of that tuning.
Below is a table showing the degrees of 31edo
which form blackjack:

Below are two pitch-height graphs showing the 31edo
version of blackjack. The graph on the left has
the "octave" divided into 12 steps, and that on the
right has it divided into 31 steps.

In the 31edo version of blackjack, L
is 2 degrees of 31edo and s is 1 degree, so L = 2s.

Below is a 5-limit
bingo-card-lattice
of 31edo, with the 21-tone blackjack scale shown in buff in the
central part of the blackjack chain which passes thru n0
and in pink in the chains which are
commatic equivalents.
The central periodicity-block
contains most of the blackjack scale, with 7 of the notes falling into
commatically-equivalent chains.
The general southwest-to-northeast trend of blackjack is obvious;
compare to the 72edo lattice above.

Below are two pitch-height graphs showing the 41edo
version of blackjack. The graph on the left has
the "octave" divided into 12 steps, and that on the
right has it divided into 41 steps.

In the 41edo version of blackjack, L
is 3 degrees of 41edo and s is 1 degree, so L = 3s.